Universidade Federal de Viçosa
Programa de pós-graduação em Genética e Melhoramento
Departamento de Biologia Geral
A new look on the genotype-by-environment interaction: enviromics and probabilistic models
Prof. Dr. Kaio Olimpio das Graças Dias
Dr. Saulo Fabrício da Silva Chaves
Background
Concepts and biological aspects
data |> ggplot(aes(y = reorder(Hybrid,GY), x = reorder(Location,GY), fill = GY)) +
geom_tile() + theme_minimal()+
scale_fill_viridis_c(option = "turbo") +
labs(y = 'Hybrid', x = 'Location')
Bayesian Inference
Bayesian Inference
Bayesian inference is reallocation of credibility across possibilities
The possibilities to which we assign credibility (probability) are the parameter values within meaningful mathematical models
Probabilities
- A Muerder Mystery (Credits: Christopher Bishop)
- A fiendish murder has been committed
- Who did it?
- There are two suspects:
- The Butler
- The cook
- There are three possible murder weapons:
- a butcher’s Knife
- a Pistol
- a fireplace Poker
Conditional Distribution
- Butler is ex-army, keeps a gun in a locked drawer
- Cook has access to lots of knives
- Butler is older and getting frail
\[ P(Weapon | Culprint) \]
Joint Distribution
- What is the probability that the Cook committed the murder using the Pistol?
\[ P(Culprint = Cook) = 80\% \]
\[ P(Weapon = Pistol | Culprint = Cook) = 5\% \]
\[ P(Weapon = Pistol, Culprint = Cook) = 80\% \times 5\% = 4\% \]
- Likewise for the other five combinations of Culprit and Weapon
\[ P(Weapon, Culprint) = P(Weapon|Culprint) \times P(Culprint) \]
PRODUCT RULE
\[ P(x, y) = P(y|x) \times p(x) \]
Generative Viewpoint
Marginal Distribution of Culprit
SUM RULE
\[ P(x) = \sum_{y}^{} P(x,y) \]
Posterior Distribution
- We discover a Pistol at the scene of the crime
Cook = 20%
Butler = 80%
Generative viewpoint
Probabilistic Models
- Samples from the posterior distributions: a simulated trial
- Let \(\Omega\) be the subset of selected candidates
- The size of \(\Omega\) is defined by the selection intensity
- After each sampling, ask:
- Which candidates are the top performers?
- Which candidates are the most stable?
- Is candidate x better than candidate y?
- Fitting a Bayesian model
\[ y_{jklm} \sim N(E[y_{jklm}], \sigma)\]
\[ E[y_{jklm}] = \mu + g_{j} + e_{k} + r_{m(k)} + b_{l(mr)} + (ge)_{jk} \]
\[ \mu \sim N (0, S^{[\mu]}) \]
\[ g_{j} \sim N (0, S^{[g]}) \]
\[ S^{\mu} \sim HalfCauchy(0, \phi) \]
Probability of superior performance
Which candidates are the top performers?
What is the risk of recommending a given candidate (performance)?
\[ Pr(\hat{g}_j \in \Omega \vert y) = \frac{1}{S} \sum_{s=1}^S I(\hat{g}_j^{(s)} \in \Omega \vert y) \]
\[ \begin{cases} \hat{g}_j \in \Omega \rightarrow I(\hat{g}_j^{(s)} \in \Omega \vert y) = 1 \\ \hat{g}_j \notin \Omega \rightarrow I(\hat{g}_j^{(s)} \in \Omega \vert y) = 0 \end{cases} \]
\(S = (\hat{g}_j \in \Omega) + (\hat{g}_j \notin \Omega)\)
- Selection for increasing or decreasing the trait value
- Which candidates are the most stable?
- What is the risk of recommending a given candidate (stability)?
\[ Pr\left[ Var(\widehat{ge}_{jk}) \in \Omega \vert y \right] = \frac{1}{S} \sum_{s=1}^S I \left[ Var(\widehat{ge}_{jk}^{(s)}) \in \Omega \vert y \right] \]
\[ \begin{cases} Var(\widehat{ge}_{jk}) \in \Omega \rightarrow I \left[ Var(\widehat{ge}_{jk}^{(s)}) \in \Omega \vert y \right] = 1 \\ Var(\widehat{ge}_{jk}) \notin \Omega \rightarrow I \left[ Var(\widehat{ge}_{jk}^{(s)}) \in \Omega \vert y \right] = 0 \end{cases} \]
- The lower, the better: stability as invariance
Pairwise probability of superior performance/stability
Is candidate x better than candidate y?
What is the probability that x performs better than y in the TPE?
\[ Pr(\hat{g}_{j} > \hat{g}_{j^\prime} \vert y) = \frac{1}{S} \sum_{s=1}^S I(\hat{g}_{j}^{(s)} > \hat{g}_{j^\prime}^{(s)} \vert y) \]
\[ \begin{cases} \hat{g}_{j} > \hat{g}_{j^\prime} \rightarrow I(\hat{g}_{j}^{(s)} > \hat{g}_{j^\prime}^{(s)} \vert y) = 1 \\ \hat{g}_{j} < \hat{g}_{j^\prime} \rightarrow I(\hat{g}_{j}^{(s)} > \hat{g}_{j^\prime}^{(s)} \vert y) = 0 \end{cases} \]
- Same comparisons using \(Var(\widehat{ge}_{jk})\)
- Marginal probabilities and conditional probabilities
- Joint probability of superior performance and stability
\[ Pr(\hat{g}_j \in \Omega \vert y) \times Pr (Var(\widehat{ge}_{jk}) \in \Omega \vert y)\]
- Diagnostics
- \(\hat{R} \rightarrow\) mean potential scale reduction factor (\(\sim 1\))
- \(WAIC2 \rightarrow\) the lower, the better
- Bayesian p-values \(\rightarrow \sim 0.5\)
- Performance/stability
- Pairwise comparison
- Probabilities within Environments
Tip
ProbBreedis a free package, available from CRAN- More information is available at the package’s vignette